Calculus Examples

$lim_{x \to 8} 3 x - \sqrt{9 x^{2} - 4 x + 4}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$lim_{x \to 8} 3 x - lim_{x \to 8} \sqrt{9 x^{2} - 4 x + 4}$

Step 2

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$3 lim_{x \to 8} x - lim_{x \to 8} \sqrt{9 x^{2} - 4 x + 4}$

Step 3

Move the limit under the radical sign.

$3 lim_{x \to 8} x - \sqrt{lim_{x \to 8} 9 x^{2} - 4 x + 4}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$3 lim_{x \to 8} x - \sqrt{lim_{x \to 8} 9 x^{2} - lim_{x \to 8} 4 x + lim_{x \to 8} 4}$

Step 5

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$3 lim_{x \to 8} x - \sqrt{9 lim_{x \to 8} x^{2} - lim_{x \to 8} 4 x + lim_{x \to 8} 4}$

Step 6

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$3 lim_{x \to 8} x - \sqrt{9 {(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 4 x + lim_{x \to 8} 4}$

Step 7

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$3 lim_{x \to 8} x - \sqrt{9 {(lim_{x \to 8} x)}^{2} - 4 lim_{x \to 8} x + lim_{x \to 8} 4}$

Step 8

Evaluate the limit of $4$ which is constant as $x$ approaches $8$ .

$3 lim_{x \to 8} x - \sqrt{9 {(lim_{x \to 8} x)}^{2} - 4 lim_{x \to 8} x + 4}$

Step 9

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$3 \cdot 8 - \sqrt{9 {(lim_{x \to 8} x)}^{2} - 4 lim_{x \to 8} x + 4}$

Step 9.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$3 \cdot 8 - \sqrt{9 \cdot 8^{2} - 4 lim_{x \to 8} x + 4}$

Step 9.3

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$3 \cdot 8 - \sqrt{9 \cdot 8^{2} - 4 \cdot 8 + 4}$

Step 10

Simplify each term.

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Step 10.1

Multiply $3$ by $8$ .

$24 - \sqrt{9 \cdot 8^{2} - 4 \cdot 8 + 4}$

Step 10.2

Raise $8$ to the power of $2$ .

$24 - \sqrt{9 \cdot 64 - 4 \cdot 8 + 4}$

Step 10.3

Multiply $9$ by $64$ .

$24 - \sqrt{576 - 4 \cdot 8 + 4}$

Step 10.4

Multiply $- 4$ by $8$ .

$24 - \sqrt{576 - 32 + 4}$

Step 10.5

Subtract $32$ from $576$ .

$24 - \sqrt{544 + 4}$

Step 10.6

Add $544$ and $4$ .

$24 - \sqrt{548}$

Step 10.7

Rewrite $548$ as $2^{2} \cdot 137$ .

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Step 10.7.1

Factor $4$ out of $548$ .

$24 - \sqrt{4 (137)}$

Step 10.7.2

Rewrite $4$ as $2^{2}$ .

$24 - \sqrt{2^{2} \cdot 137}$

Step 10.8

Pull terms out from under the radical.

$24 - (2 \sqrt{137})$

Step 10.9

Multiply $2$ by $- 1$ .

$24 - 2 \sqrt{137}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$24 - 2 \sqrt{137}$

Decimal Form:

$0.59060017 \dots$